Surfaces with central convex cross-sections (Q424802)

The author studies surfaces in the 3-dimensional Euclidean space. A surface is said to have the central plane property if and only if: {\parindent=6mm\begin{itemize}\item[-] The surface \(S\) intersects at least one affine plane transversally along an oval. \item[-] Every time \(S\) intersects an affine plane transversally along an oval, that oval is central. NEWLINENEWLINE\end{itemize}} The authors show that a complete immersed surface with this property must be either a cylinder or a quadric. This result is inspired by an old result of Blaschke form 1918 which characterises quadrics as the smooth convex surfaces for which every nearly tangent tranverse plane cuts the surface along a central loop.NEWLINENEWLINEHowever the method used is completely different from the one used by Blaschke as it also allows to deal with the negative curvature case. As such they also characterise the one sheeted hyperboloids as the only negatively curved surfaces that contain a tranverse plane oval but no skewloop.